2007 trial papers

8/10/2019 2007 Trial Papers

http://slidepdf.com/reader/full/2007-trial-papers 1/40

Friday 21 September: 7:00 a.m .Time: 3 hours 10 minutes

Examination material: one 41– page question booklet

Approved dictionaries, notes, calculators and computer software may be used

Instructions to Candidates

1. You will have 10 minutes to rea the paper. You must not write in !our question booklet or use a"al"ulator uring this rea ing time but !ou ma! make notes on the s"ribbling paper provi e .

#. $nswer all parts o% &uestions 1 to 1' in the spa"es provi e in this question booklet. There is no nee

to %ill all the spa"es provi e . You ma! write on pages #1( 3) an 40 i% !ou nee more spa"e( makingsure to label ea"h answer "learl!.3. The total mark is approximatel! 14'. The allo"ation o% marks is shown below:

&uestion 1 # 3 4 * ' + , ) 10 11 1# 13 14 1* 1'

-arks 10 ' ) * 11 * 11 ) + ' 10 ) 13 14 1# )

4. $ppropriate steps o% logi" an "orre"t answers are require %or %ull marks.*. how all working in this booklet. /You are strongl! a vise not to use s"ribbling paper. ork that

!ou "onsi er in"orre"t shoul be "rosse out with a single line.'. 2se onl! bla"k or blue pens %or all work other than graphs an iagrams( %or whi"h !ou ma! use a

sharp ark pen"il.+. tate all answers "orre"t to three signi%i"ant %igures( unless otherwise state or as appropriate.,. iagrams( where given( are not ne"essaril! rawn to s"ale.). The list o% mathemati"al %ormulae is on page 41. You ma! remove this page %rom the booklet be%ore

the examination begins.10. rite !our name( stu ent number an group in the spa"e provi e at the top o% this page.11. omplete the box on the top right5han si e o% this page with in%ormation about the ele"troni"

te"hnolog! !ou are using in this examination.

6ages: 41&uestions: 1'

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tu ent 7o : 8888888888888888888888888888

9roup : 88888888888888888888888888

9raphi"s al"ulator

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-o el 88888888888888

omputer o%tware

TRIAL EXAMINATI N 2007 MAT!EMATI"AL ST#$IES



%LAN& %LAN&

#



/" ;in dx x

ex x∫

−+−

141

3# 3 .

/3 marks

/ Evaluate ∫ −

+−0

#

13 #

dxe x

.

/1 mark

4



'#ESTI N 2

Vitality is one o% the bran s o% an organi" tomato <ui"e rink. =ts pro u"er "laims that #0> o% all organi"tomato <ui"e rinkers pre%er Vitality . To test the vali it! o% this "laim( a "ompetitor samples #*0 organi"tomato <ui"e rinkers an %in s that onl! 33 pre%er the Vitality bran .

/a ase on this sample( %in a )*> "on%i en"e interval %or p( the proportion o% all organi" tomato <ui"erinkers who pre%er the Vitality bran . =nterpret !our answer.

/# marks

/b oes the sample support the pro u"er?s "laim@ Explain !our answer.

/# marks

/" =% the "ompetitor wishes to estimate( with a probabilit! o% 0.)*( the proportion to within #>( howlarge shoul the sample siAe be@

/# marks

*



'#ESTI N (

/a =%

−−=

p A

*

31( %in the values o% p an q su"h that qI A A += 3# ( where p an q are s"alars an

I is the # x # i entit! matrix.

/3 marks

/b Ben"e %in 1−

A in linear %orm.

/3 marks

'



/" 9iven that

=

413

*,C ( %in matrix B su"h that ( ) I C I B A 10* +=− using matrix algebra.

/3 marks

+



'#ESTI N )

$ tennis a"a em! in $ elai e trains !oung pla!ers who aspire to turn pro%essional whilst still in theirteens. Cne o% the areas o% the game that this a"a em! %o"uses on is the a"e serve.

$n a"e is a serve that goes untou"he b! the serverDs opponentDs ra"ket an thus is unreturne .

e"or s show that the <unior trainees in this a"a em! "an serve an a"e 3*> o% the time.

/a Barr!( a <unior trainee( atten s a morning training session where he makes twelve serves. hat is the probabilit! that

/i none o% the serves is an a"e@

/1 mark

/ii between 3 an + /in"lusive a"es are serve @

/# marks

,

http://en.wikipedia.org/wiki/Serve_(tennis)

http://en.wikipedia.org/wiki/Racket

http://en.wikipedia.org/wiki/Serve_(tennis)

http://en.wikipedia.org/wiki/Racket



/b ! the en o% the a!( Barr! has ma e a total o% 1*0 serves. ;in the probabilit! that he serve an a"eat least hal% o% the time.

/# marks

)



'#ESTI N *

onsi er the %ollowing s!stem o% equations:( ) *#1

##

1#3

# −=−−+−=++=+−

k kz yk x

z ky x

z y x

( where k is real.

/a how that this s!stem "an be re u"e to −+−−

1

4#

1

030

#40

#31

# k

k

-k k .

/4 marks

/b tate the value o% k %or whi"h the s!stem has no solution.

/1 mark

10



/" /i tate the value o% k whi"h gives an in%inite number o% solutions.

/1 mark

/ii 9ive these solutions in parametri" %orm.

/# marks

11



/ /i tate the value o% k %or whi"h the s!stem has unique solution.

/1 mark

/ii olve the s!stem when 1−=k .

/# marks

1#



'#ESTI N +

The graphs o% the %un"tions ( ) x f y = an ( ) x f y ′= are shown in the iagram below( where e x ≤≤0 .

9raph I has a turning point at A an "hanges shape at D . 9raph II has a turning point at B an "hangesshape at C .

/a tate whi"h graphs represent ( ) x f an ( ) x f ′ respe"tivel!.

/# marks

/b Cn the same set o% axes above( sket"h the %un"tion ( ) x f y ′′= . /3 marks

A B

D

C

9raph I9raph II

x

y

Oe

13



'#ESTI N 7

$n experimental "ar that is %uelle b! peanut oil is taken %or several test runs in a remote part o% $ustralia.Cn one o% its test runs( it was seen to be traveling in a straight line with initial velo"it! o% 1ms3 −− . =tsa""eleration t se"on s a%ter motion began was ( ) 5#ms#4 t t a −= ( where t is time in se"on s an 0≥t .

/a ;in the velo"it! %un"tion ( )t v an raw its sign iagram.

/4 marks

/b how that the "ar was stationar! a%ter 1 se"on . $t what other time was the "ar at rest@

/# marks

14



/" ith the help o% a motion iagram( "al"ulate the istan"e travelle b! this experimental "ar in the %irst* se"on s o% its motion.

/* marks

1*



'#ESTI N ,

onsi er the %ollowing number gri in whi"h ea"h "ell "ontains a single whole number:

a b ! "a b ! #a

a ! c a a ! #c

b ! $a b a ! c

This number gri is name the %A&doku an amongst its unique %eatures is the %a"t that the sum o% thenumbers in ea"h row is equal to the same value M ( the sum o% the numbers in both the iagonals is equalto another value N ( an that all the numbers %rom ea"h "ell a up to 4*.

/a 2sing the variables a( b an c, write own a s!stem o% 3 equations in 3 unknowns that was extra"te%rom the %A&doku gri shown above.

/* marks

1'



/b /i Ben"e %in the values o%a( b an c'

/# marks

/ii omplete the %A&doku below with !our answers %rom part /b /i above.

the %A&doku

/# marks

1+



'#ESTI N -

-!oglobin is a protein %oun in the mus"le "ells o% animals. =t %orms pigments responsible %or makingmeat re . $ laborator! anal!ses spe"imens o% m!oglobin taken %rom %owls %oun in a "ertain area to

etermine the "on"entration o% an a"tive ingre ient( i enti%ie onl! as -9F51. The "on"entrations areassume to %ollow a normal istribution. The stan ar eviation is a propert! o% the anal!ti"al pro"e urean is known to be 0.0', miligrams.

The laborator! anal!ses a sample o% twent! spe"imens o% m!oglobin an reports that the mean result o% the twent! rea ings is ,40,.0= x miligrams.

$ s"ientist emplo!e at the laborator! is aske to evaluate( using a two5taile ( 5test at the *> level o% signi%i"an"e( the "laim that the "on"entration o% the a"tive ingre ient is 0.,' miligrams.

/a tate the null h!pothesis.

/1 mark

/b tate the alternative h!pothesis.

/1 mark

/" tate the null istribution o% the test statisti".

/1 mark

1,



/ etermine whether or not the null h!pothesis shoul be re<e"te .

/3 marks

/e hat "an we "on"lu e %rom the results in / @

/1 mark

1)



*ou may write in t+is section if you need additional space' Clearly number eac+ question and partattempted in t+is section for example, -uestion . c/ i//'

#1



'#ESTI N 11

onsi er the "urve ( ) y xy x )33# #3 −=−− .

/a how that( )

)'

3#3 ##

−

−−= xy

y x

dx

dy.

/4 marks

##



/b ;in the equation o% the normal to the "urve at ( )1(# .

/3 marks

/" 9iven that the "urve has a horiAontal tangent at ( )ba ( ( %in b in terms o% a'

/3 marks

#3



'#ESTI N 12

e%orestation is one o% the ma<or problems %a"ing sub5 aharan $%ri"a.$lthough the "learing o% lan %or %arminghas been the ma<or "ause( the stea il!in"reasing eman %or %uel woo has

be"ome a signi%i"ant %a"tor.

This graph shows the rates o% %uel woo"onsumption an new tree growth /both inmillions o% "ubi" meters per !ear in

u an %rom 1),0 to #000( taken %rom there"or s o% the orl ank.

uppose that the rate o% %uel woo "onsumption /in millions o% "ubi" meters per !ear in u an t !earsa%ter 1),0 is given approximatel! b! the %un"tion ( ) t et C 03.0#.+'= ( where 0≥t .

/a ;in / t C ′ .

/# marks

/b how that / t C ′ G 0 %or all values o%t . hat is the signi%i"an"e o% this result@

/# marks

1),0 1),* #0001))*1))0

1*0

100

*0

ate o% %uel woo "onsumption

ate o% new tree growth

#4



/" etermine the amount o% %uel woo that will be "onsume %rom 1),0 to #000.

/# marks

The rate o% new tree growth /in millions o% "ubi" meters per !ear in the u an t !ears a%ter 1),0 is given

approximatel! b! the %un"tion ( ) t

et 0 0).0

03.'*0−=

.

/e rite own an expression to represent the amount o% epletion o% the %orests ue to ex"ess %uel woo"onsumption over new growth %rom 1),0 to #000.

/# marks

/% Ben"e( etermine the amount o% epletion o% the %orests %rom 1),0 to #000.

/1 mark

#*



'#ESTI N 1(

6inot 7oir is one o% the ol est grape varietiesto be "ultivate %or the purpose o% makingwine. $ vine!ar uses mainl! 6inot 7oir to

pro u"e its premier bottles o% re wine %or export. These bottles are kept in a "ool "ellar.

The temperature o% the bottles o% 6inot 7oir isa normall! istribute variable 1 ( with a mean10.*4H an a stan ar eviation o% 1.0,H .= eall! the temperature o% these bottles o% wine shoul be kept at a level o% no more than13.0 H .

/a ;in the probabilit! that a ran oml! sele"te bottle o% 6inot 7oir %rom the "ellar will have atemperature o% no more than 13.0 H .

/1 mark

/b ine6ress( a wine export "ompan!( bu!s 1*00 bottles o% 6inot 7oir %rom this vine!ar .$pproximatel! how man! bottles will have a temperature o% more than 13.0 H @

/# marks

#'



The manager measures the temperature o% * bottles o% 6inot 7oir ever! a! an re"or s the meantemperature as *

2 .;or pro u"t "ontrol purposes( the manager also measures the temperature o% *0 bottles o% 6inot 7oir anre"or s the mean temperature( *0

2 .

Two histograms are shown below. Cne illustrates the istribution o% * 2

an the other illustrates theistribution o% *0

2 .

/" hi"h histogram illustrates the istribution o% *0 2 @ Explain !our answer.

/# marks

!i to/ram A

,.* 9.5 10.5 11.5 12.5

Temperature ( 1 /o

!i to/ram %

8.5 9.5 10.5 11.5 12.5

Temperature( 1 /o

#+



The bottles o% 6inot 7oir are pa"ke into "rates o% #* bottles ea"h. uppose that #* 2 represents the mean

temperature o% #* bottles o% 6inot 7oir in a "rate.

/ hat "an we sa! about the shape o% the sampling istribution o% #* 2 @

/1 mark

/e how that the istribution o% #* 2 has a mean temperature o% 10.*4H an stan ar eviation o%

0.#1'H .

/# marks

/% etermine the proportion o% "rates that will "ontain a mean temperature greater than 10.,*H .

/1 mark

#,



The manager is "on"erne that ue to "hanging weather patterns( the mean temperature o% the bottles o% 6inot 7oir kept in ea"h "rate in the "ellar is not at its i eal temperature o% no more than 13.0 H . =% so( hewill nee to a <ust the temperature "ontrol in the "ellar. To estimate µ ( the mean temperature o% all

bottles o% 6inot 7oir in the "ellar( he sele"te a "rate o% #* bottles at ran om an %oun that the meantemperature is 1#.4,H ( with a stan ar eviation o% 1.0,H .

/g al"ulate a )*> "on%i en"e interval %or µ .

/# marks

/h ase on this "on%i en"e interval( shoul the manager a <ust the temperature "ontrol in the "ellar@Iusti%! !our answer.

/# marks

#)



The graphs o% ( ) ( ) x

x x+

3ln= an ( ) x

x ) #

1= where 0> x ( are shown below.

/b how algebrai"all! that ( ) ( ) x ) x+ = when3e

x = .

/3 marks

a

( ) ( ) x

x x+

3ln=

y

x

( ) x

x ) #1

=

31



/" 9iven that x 3 a is the point where ( ) x+ y = "uts the x5axis( %in a.

/# marks

/ /i rite own the "al"ulus expression that represents the area that is en"lose b! the graphs o% ( ) x+ y = ( ( ) x ) y = and the x5axis %rom x 3 a to x 3 *. ha e this area on the graph given above.

/# marks

/ii Ben"e( %in the exa"t area that is en"lose b! the graphs o% ( ) x+ y = ( ( ) x ) y = and the x5axis%rom x 3 a to x 3 *.

/3 marks

3#



Question 15 starts on page 34

33



/" ;in the value/s o% t %or whi"h the number o% unemplo!e people was in"reasing.

/3 marks

/ =n whi"h month an !ear i this "ountr! have the highest number o% unemplo!e people@ Ben"estate the number o% unemplo!e people.

/3 marks

/e hen was the number o% unemplo!e people in"reasing at the greatest rate@

/1 mark

3*



'#ESTI N 1+

$t the 2niversit! o% ollongong( the "hool o% -usi" an rama is sel%5sustaining( that is( it raises itsown %un s to support its ail! running o% a%%airs. Cne o% its %un raising "hannels is its in5house ineplex(whi"h o%%ers movies at a%%or able ti"ket pri"es an is open to all universit! stu ents.

The movies s"reene at the ineplex are ivi e into three genres: "ome !( horror an rama. Thetransition iagram below illustrates the probabilit! that a universit! stu ent who wat"hes a movie %romone genre will wat"h a movie %rom another /or the same genre the %ollowing week.

/a omplete the transition matrix T below base on the given iagram:

=drama

+orror

comedy

from

drama+orror comedy

to

T

/# marks

Comedy

Horror

Drama

0.#

0.'

0.4

0.*

0.3

0.4

3'



/b Cn 1 Ian #00'( "ome ies ha a market share o% 3#>( the horror movie genre ominate the marketwith its share o% *1>( an ramas hel their own with a 1+> market share.

onstru"t a 1 x 3 matrix S whi"h illustrates the market shares o% "ome !( horror an rama moviesrespe"tivel!.

/1 mark

/" hat market share i horror movies hol a%ter /i the %irst week@

/1 mark

/ii three weeks@

/1 mark

3+



/ hat is the soonest time /in weeks that these three movie genres a"hieve stea ! state proportions inthe market share@ rite own the matrix that shows the proportions o% market share hel b! thesethree movie genres in the long run.

/# marks

/e $ssume that the number o% universit! stu ents who patroniAe the ineplex ever! week remaine"onstant. =% it was %oun that in the long run 1*# universit! stu ents wat"he a "ome ! ever! week(how man! stu ents in total patroniAe the ineplex@

/# marks

3,




3)

2007 trial papers

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